A first aim of this paper is to give a kind of counterpart to [1]}, where left cancellative left semi-braces having the set of idempotents as an ideal were studied, considering left cancellative left semi-braces \((B,+,\circ )\)  for which the skew left brace \(G\)  contained in \(B\)  is the ideal given by \(G:=Ker(\lambda _{|_E})\) , i.e. \(G\)  coincides with the kernel of the action on the set of additive idempotents induced by the map \(\lambda \) . We will show that these left semi-braces come from a skew left brace \(G\)  and a groups semidirect product \(G\rtimes _{\beta } H\)  where \(\beta \)  acts on the skew left brace \(G\) . As a main application, we will use these left semi-braces to give a complete classification of left cancellative left semi-braces having size \(pq\)  (for arbitrary prime numbers \(p,q\) ) and \(2p^2\)  (for an arbitrary prime number \(p>2\) ) such that the set of additive idempotents \(E\)  is a Sylow subgroup of \((B,\circ )\) . Since \(E\)  always is a subgroup of \((B,\circ )\)  and left cancellative left semi-braces with \(|E|=1\)  are nothing more than skew left braces, our result, together with the main Theorem of [2]}, gives a complete classification of left cancellative left semi-braces of size \(pq\) .
